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Iterative seislet thresholding

When $ \mathbf{S}=\mathbf{A}^{-1}\mathbf{T}_{\tau}\mathbf{A}$ , the iterative framework refers to the iterative seislet thresholding:

$\displaystyle \mathbf{m}_{n+1} = \mathbf{A}^{-1}\mathbf{T}_{\tau_n}\mathbf{A}[\mathbf{m}_n+\lambda\Gamma^*[\mathbf{d}-\Gamma\mathbf{m}_n]],$ (3)

where $ \mathbf{T}_{\tau_n}$ denotes a thresholding operator with a threshold $ \tau_n$ , $ \mathbf{A}$ and $ \mathbf{A}^{-1}$ are a pair of forward and inverse seislet transforms.

$ \mathbf{T}_{\tau}$ can be either a soft-thresholding operator:

$\displaystyle \mathbf{T}^s_{\tau}(x) = \left\{ \begin{array}{ll} (\vert x\vert-...
...vert x\vert\ge\tau  0 &\text{for} \quad \vert x\vert<\tau \end{array}\right.,$ (4)

or a hard thresholding operator:

$\displaystyle \mathbf{T}^h_{\tau}(x) = \left\{ \begin{array}{ll} x &\text{for} ...
...ert x\vert\ge\tau  0 &\text{for} \quad \vert x\vert<\tau \end{array}\right. .$ (5)

In this paper, all the examples are based on soft thresholding. I iteratively decrease the threshold $ \tau$ in the seislet domain during the iterations.